clear, close, clc

N = 10000;

a = [1, -1.5, 0.7];
b = [0.0, 1.0, 0.5];
c = [1.0, -1.0, 0.2];

B_A = tf(b, a, 1.0);
C_A = tf(c, a, 1.0);

e = randn(N, 1);
e_u = randn(N, 1);

u = generate_input_sequence(e, e_u);

y = lsim(B_A, u) + lsim(C_A, u);

%% Exercise 8.1

C_inv = tf([1, 0, 0], c, 1.0);

u_F = lsim(C_inv, u);
y_F = lsim(C_inv, y);

phi_u_F = toeplitz(u_F(1:end-1), zeros(2, 1));
phi_y_F = toeplitz(-y_F(1:end-1), zeros(2, 1));
phi = [phi_y_F, phi_u_F];

theta = phi \ y_F(2:end)
% G_arx = arx([y_F u_F], [2 2 1])

%% Exercise 8.2

N_val = 1000;
y_val = y(end-N_val:end);
u_val = u(end-N_val:end);

G_pred = tf([theta(3), theta(4)], [1.0, theta(1), theta(2)], 1.0);
y_pred = lsim(G_pred, u_val);

plot(y_val, '--black');
title('Validation plot')
xlabel('Time')
ylabel('Output')
hold on
plot(y_pred, 'r')

%% Exercise 8.3 is too ez and I'm mega lazy

%% Helper functions
function u=generate_input_sequence(e, e_u)
    N = size(e, 1);
    u = zeros(N, 1);

    for i=1:N
        if i > 1
            u(i) = u(i) + 0.1 * u(i - 1);
        end
        if i > 2
            u(i) = u(i) + 0.12 * u(i - 2);
        end

        u(i) = u(i) + e(i) + e_u(i);
    end
end